Shear stress concentration

High shear stress concentration in the matrix near the fiber ends. 

Stresses can can be concentrated by various mechanisms. Perhaps the most simple of these is the one used by Zener (1946) to explain the grain size dependence of the yield stresses of polycrystals. This is the case of the shear crack which was studied by Inglis (1913). Consider a penny-shaped plane region in an elastic material of diameter, D, on which slip occurs freely and which has a radius of curvature, p at its edge. Then the shear stress concentration factor at its edge will be = (D/p)1/2.The shear stress needed to cause plastic shear is given by a proportionality constant, a times the elastic shear modulus,... 

Impurities and flaws have a detrimental effect on the fibre strength. Due to shear stress concentrations at structural irregularities and impurities, the ultimate debonding stress r0 ( rm) or the critical fracture strain / may be exceeded locally far sooner than in perfectly ordered domains. Thus, during the fracture process of real fibres the debonding from neighbouring chains occurs preferably in the most disoriented domains and presumably near impurities. At the same time, however, the chains in the rest of the fibre are kept under strain but remain bonded together up to fracture. 

Maximum absolute stress concentration factors are shown in Figure 5, again for an ellipticity R = 0.1. As before, all orientations p give the same stress concentration for equal biaxial tension ( = 45°). In addition equal biaxial compression ( = —135°) gives the same stress concentration since absolute stress concentration is being considered. The shear stress concentration is given by Q /2. The curves shown in Figure 5 are symmetrical about the pure shear condition ( = —45°). 

The modulus and yield kinetic parameters of the block polymer B can be related to those of the homopolymer in terms of a microcomposite model in which the silicone domains are assumed capable of bearing no shear load. Following Nielsen (10) we successfully applied the Halpin-Tsai equations to calculate the ratio of moduli for the two materials. This ratio of 2 is the same as the ratio of the apparent activation volumes. Our interpretation is that the silicone microdomains introduce shear stress concentrations on the micro scale that cause the polycarbonate block continuum to yield at a macroscopic stress that is half as large as that for the homopolymer. The fact that the activation energies are the same however indicates that aside from this geometric effect the rubber domains have little influence on the yield mechanism. 

The probability of crack propagation can be calculated for the classic hexagonal cell and can be related to cell dimensions and the applied stress.9 A new constant is involved in the resultant equations it is related to cell size (since crack propagation is a combination of tensile stress and bending moment). For CMP pads, the equivalent areas of concern will be the cell density (related to the relative density) and the cell size distribution (visualized in Figure 6.33). It is the cell proximity or overlap probability that will determine the material s inherent propensity to support crack propagation. Also, since much of the shear stress concentration is at the top frictional surface, the damage done to the inherent cell structure by the CMP process is likely to determine much of the material s wear response. It is probable that the shape, size, and distribution of abrasives play an important role in the material s wear characteristics. 

Wake(37) and by Adams and Wake(5), and Kinloch(4) summarises the evolution of the approach of the many stress analysts. The most common shear test comprises the single lap shear joint embodied in BS 5350(10) and ASTM 01002-72(11) (Fig. 4.7(a)). With reference to Figs. 4.1(a) and 4.8 it can be seen that the resulting stress concentrations can be extremely large at the joint ends (points X and Y in Fig. 4.8(b)). The analysis of Volkersen(15) predicts that for identical adherends, the elastic shear stress concentration factor, for the adhesive due to adherend tensile strain is given by... 

Thus decreasing the overlap length or shear modulus of the adhesive, or increasing the adherend stiffness or adhesive layer thickness, will decrease the shear stress concentration in the adhesive layer. 

Taking into account that, for polycrystals, the applied stress is one-third of the shear stress acting on the pile-up plane, the relationship between the shear stress concentration (x) and the applied stress a (x/a) must lie in the range between 40 and 75, although this would be too large to be explained by the stress concentration induced at multiple-grain junctions during GBS. Clearly, the dislocations reported by... 

A thicker bond line can reduce shear stress concentration by spreading the strain over a larger dimension, resulting in less strain on the adhesive. An increase in bond line gap from 0.0025 to 0.10 cm can decrease the stress ratio from 18.2 to 3.06. ... 

In the bonded joint design the most basic problems are the unavoidable shear stress concentrations and the inherent eccentricity of the forces causing peel stresses both in the adhesive and in the adherends. At the ends of the overlap both the peel and shear stresses reach their maximum values, resulting in reduced load-bearing capacity of the joint, see Figure 5.28. 

Fig. 9. SEM of damaged C fibres, caused by high shear stress concentration of the matrix on the fibre during the first carbonization, due to good fibre/matrix adhesion and hindered longitudinal shrinkage of the matrix precursor.

Other modifications of the lap joint include tapering of the outside or inside of the adherend at the joint. This may produce a slight increase in strength with a reduction in the shear stress concentration in the joint. 

Several methods have been proposed for predicting the stress state at the interface, which can then be used to estimate the bond strength. The shear lag method has received extensive treatment by several investigators. This method determines the interface shear stress concentration at the end of the fiber as well as shear stress variation along the fiber. Additional methods include the Lame solution for a shrink fit, classical elasticity boundary value problems, and finite-element analysis. 

The maximum shear stress concentration depends on the fiber type and fiber volume ratio. 

An ellipsoidal end yields the lowest interfacial shear stress concentration. 

The elastic nature of fiber and shear stress concentration at the fiber ends affected the tan 6 value when incorporated in a composite material, which is related to the additional viscoelastic energy dissipation in the matrix material [15], Figure 13.8 shows a selection of tan 6 curves in which it can be seen that the tan 6 peak shifted to the higher temperatures, broadened, and decreased as the kenaf whiskers content increased. The decrease of tan 6 value indicated the reduction of macromolecular mobility of the fiber surface environment, whereas better interaction between kenaf whiskers and matrix can be concluded [16]. 

Thus, the stress concentration factor, t]c, becomes simply a function of a single dimensionless coefficient, A. It can, therefore, be readily appreciated that the theory of Volkersen predicts that decreasing the overlap length, U, or shear modulus of the adhesive, Ga, or increasing the stiffness of the substrate or the thickness of the adhesive layer, will decrease the shear stress concentrations in the adhesive layer. However, whether these changes result in stronger joints... 

Adams and Peppiatt [60] have considered the problem of the in-plane transverse stresses and to ascertain the magnitude of such stresses they have used experimental models and analytical and finite-element analyses solutions of the Volkersen theory, but in three dimensions. They demonstrated that Poisson s ratio strains generated in the substrates cause shear stresses, T13, in the adhesive layer and tensile stresses, 0-33, in the substrate acting transverse to the direction of the applied load, but in the plane of the joint. For metal-to-metal joints the transverse shear stress, has a maximum value of about one-third of the maximum longitudinal shear stress, ri2(max), and this occurs at the corners of the overlap. This, therefore, enhances the shear stress concentration which exists at this point due to the effects described above. Bonding substrates of dissimilar stiffness produces greater stress concentration in the adhesive than when similar substrates are employed. 

Geometry effects - from the above discussions, and in particular Equations 6.15, 6.17 and 6.19, it is evident that the shear stress concentrations in the adhesive layer near the ends of the overlaps increase with increasing overlap length. This is directly reflected by the observation that the average applied shear stress, Tf, at joint fracture decreases with increasing overlap length, /a, as... 

Figure 6,24 Shear stress concentrations for various lap joint designs, elastic analyses for typical epoxy adhesive/aluminium alloy joints [6].

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